Most simply, the graphs of sine and cosine relate to each of the others, in how the others' asymptotes would be where either sine or cosine are equal to zero. This is because sine is the denominator for cosecant and cotangent, and cosine, the denominator for secant and tangent.
Sine is equal to zero, when on the x-axis, it is at zero, and π, as illustrated by the graph below. This is so, because sine is y/r on the unit circle, and therefore, shall only be equal to zero when y is equal to zero. And, y, on the unit circle, is equal to zero, at those places - 0° (zero) and 180° (π). Therefore, there shall be asymptotes for cosecant and cotangent when sine is equal to zero. Asymptotes are important if you want to find out why cotangent goes down. It basically goes down, because the two areas where uninterrupted, consecutive points can continue from left to right, dictate that the left area shall always be positive, and that the right area shall be negative. This is because of the unit circle's ASTC thing. So, since cotangent has to go from positive to negative, it has to go downhill. It's the only way. The only way that I could possibly see sine relating to cosecant, is that they share the same positive/negative areas of the graph. In the areas where sine is positive, cosecant is positive, and vice-versa.
Cotangent is the orange line, and cosecant is the blue line.
Cosine is equal to zero, when on the x-axis, it is at π/2 and 3π/2, as illustrated by the graph below. This is so, because cosine is x/r on the unit circle, and therefore, shall only be equal to zero when x is equal to zero. And, x, on the unit circle, is equal to zero at those places - 90° (π/2) and 270° (3π/2). Therefore, there shall be asymptotes for secant and tangent when cosine is equal to zero. Asymptotes are important if you want to find out why tangent goes up. It basically goes up, because the two areas where interrupted, consecutive points can continue from left to right, dictate that the left area shall always be negative, and that the right area shall always be positive. This is because of the unit circle's ASTC thing. So, since tangent has to go from negative to positive, it has to go downhill. The only way that I could possibly see cosine relating to secant, is that they share the same positive/negative areas of the graph. In the areas where cosine is positive, secant is positive, and vice-versa.
Tangent is the orange line, and secant is the blue line.
